Amy has a master's degree in secondary education and has taught math at a public charter high school.
Watch this video lesson to learn how you can turn your system of equations into matrix form. The matrix form is another way of writing your linear system that is sometimes easier to work with.
A Linear System
In this video lesson, we talk about the linear system, a collection of linear equations. Recall that a linear equation is an equation of degree one with only two variables, usually x and y. Our system, then, is a collection of two or more of these linear equations. We usually see our linear systems written as a list of equations with a bracket in front of them, like this:
This is the most common way you will see linear systems written. Notice that we have our x terms first, followed by our y terms, then an equals sign, and finally our constant.
An Augmented Matrix
The other form in which we can write our linear systems is called an augmented matrix, which is a combination of two matrices. If you look at just the coefficients of our linear system along with their sign, then you can split our linear system into one matrix on the left side of the equation and another matrix on the other side of the equation. Combining these two matrices then gives us our augmented matrix. We use a vertical line to note where our equals sign is that splits the matrix. So our augmented matrix looks like this:
We have only our coefficients listed. Our coefficients are organized into their appropriate columns. All the x term coefficients are in the first column, all the y term coefficients are in the second column, and all our constants are in the final column. The vertical lines show you where the equals sign is. We can easily write our linear equations directly on top of our matrix numbers, and you can see how the numbers line up nicely with each other:
The linear equations line up with the matrix numbers.
Writing an augmented matrix from a linear system is easy. First, you organize your linear equations so that your x terms are first, followed by your y terms, then your equals sign, and finally your constant. Once you have all your equations in this format, all it takes for you to turn it into an augmented matrix is to write your coefficients down along with their positive or negative sign. For the equals sign, you use a vertical line.
Let's look at how to turn this linear system into an augmented matrix.
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We first make sure that all of our equations are organized so that the x terms are first, followed by the y terms, the equals sign, and then the constant. We look at our equations, and we see that they are already in this format. We can now go ahead and isolate all the coefficients. For the first row, we have 7, -8, and 1. The second row we have -4, 3, and 0. The third row, we have 0, 4, and 9. We have 0 first because there is no x term, so the number is 0. Now we can put all these numbers in order in our augmented matrix using a vertical line for the equals sign.
And we are done!
Let's try another one. Let's write this linear system into an augmented matrix.
Looking at our equations, we see that we need to rewrite our second equation so that the x term comes first. We rewrite it to 3x + 2y = 4. Now we can isolate the coefficients. The first row has 1, -1, and 2. The second row has 2, 3, and 4. Plugging these into our augmented matrix, we get this:
And we are done!
It is fairly easy and straightforward to turn a linear system into an augmented matrix. Let's review what we've learned. We've learned that a linear system is a collection of linear equations. A linear equation is an equation with two coefficients and no exponents. A system will have at least two equations.
An augmented matrix is a combination of two matrices, and it is another way we can write our linear system. When written this way, the linear system is sometimes easier to work with. To write our linear system in augmented matrix form, we first make sure that our equations are written with the x term first, followed by the y term, then the equals sign, and finally the constant. We then isolate our coefficients along with their signs and put them in matrix form in the order in which they appear, row by row. We use a vertical line to stand for the equals sign.
After this lesson, you'll be able to:
Define linear system and augmented matrix
Explain how to turn a linear system into an augmented matrix form
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